Interplays between Harmonic Analysis and Inverse Problems

Objective:

This scientific proposal addresses research in the fields of harmonic analysis and inverse problems, and explores certain interactions. The suggested plan of investigation in these areas is natural as there is a high degree of commonality in the tools required to address the problems in hand, which vary from purely theoretical questions in harmonic analysis to more concrete ones in the theory of inverse problems. In harmonic analysis we are interested in research related to Kakeya and restriction phenomena. In the first case, questions are motivated by determining a certain notion of dimension, the Hausdorff dimension, of sets that contain a unit line segment in every direction. The study of these special sets has a long history and is connected to a host of central questions and conjectures in harmonic analysis. Our point of view is that of the analysis of suitable directional operators, namely, we consider averaging operators whose mapping properties capture the behaviour of these sets. The study of these operators, as well as of the closely related directional singular integrals, is also motivated by the problem of differentiation of functions along directions in a given set, or even along directions dictated by a suitable vector field. Again these objects are related to some long-standing conjectures in harmonic analysis and even conditional results are known to imply, for example, the Carleson theorem on the convergence of partial sums of Fourier series. The approach we suggest is in line with recent developments in harmonic analysis and is strongly influenced by the polynomial method. Restriction phenomena refer to the possibility of restricting the Fourier transform of a function to hypersurfaces with curvature. Such results can be used to deduce fine convergence properties of solutions of equations, like the Schrödinger equation, or wave equations. Our interest is in an appropriate notion of size of the exceptional sets, namely the sets where these convergence results fail. In inverse problems we are interested in questions of regularity for certain inverse boundary value and scattering problems. The main goal in the field of inverse problems is to model some remote sensing strategies, non-destructive testing, or medical imaging techniques, to contribute in their developments, and propose possible improvements. In mathematical terms the aim is to determine the coefficients of the equations describing a specific phenomenon, given some admissible knowledge of their solutions. Typically, the coefficients represent the medium and the information on the solutions is only accessible through non-invasive measurements. We propose the study of three different problems. The first one is usually called the Calderón problem and we intend to determine a bounded conductivity with unbounded gradient from boundary values of the corresponding solutions. The second problem arises in the context of heat transfer in a fluid and our goal is to determine a bounded advection coefficient. The objective of the third problem is to show that the scattering data are enough to determine electric potentials containing specific singularities on points and surfaces.